Open Access

Oscillation of Even-Order Neutral Delay Differential Equations

Advances in Difference Equations20102010:184180

DOI: 10.1155/2010/184180

Received: 28 November 2009

Accepted: 29 March 2010

Published: 19 May 2010

Abstract

By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq2_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq3_HTML.gif is even, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq5_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq6_HTML.gif . These oscillation criteria, at least in some sense, complement and improve those of Zafer (1998) and Zhang et al. (2010). An example is considered to illustrate the main results.

1. Introduction

This paper is concerned with the oscillatory behavior of the even-order neutral delay differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq7_HTML.gif is even https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq8_HTML.gif

In what follows we assume that

(I1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq10_HTML.gif ,

(I2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq17_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq18_HTML.gif is a constant,

(I3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq21_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq22_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq23_HTML.gif is a constant.

Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale [1].

In the last decades, there are many studies that have been made on the oscillatory behavior of solutions of differential equations [26] and neutral delay differential equations [723].

For instance, Grammatikopoulos et al. [10] examined the oscillation of second-order neutral delay differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq24_HTML.gif

Liu and Bai [13] investigated the second-order neutral differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq25_HTML.gif

Meng and Xu [14] studied the oscillation of even-order neutral differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq26_HTML.gif

Ye and Xu [21] considered the second-order quasilinear neutral delay differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq27_HTML.gif

Zafer [22] discussed oscillation criteria for the equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq28_HTML.gif

In 2009, Zhang et al. [23] considered the oscillation of the even-order nonlinear neutral differential equation (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq29_HTML.gif

To the best of our knowledge, the above oscillation results cannot be applied when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq30_HTML.gif and it seems to have few oscillation results for (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq31_HTML.gif

Xu and Xia [17] established a new oscillation criteria for the second-order neutral differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ7_HTML.gif
(1.7)

Motivated by Liu and Bai [13], we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq32_HTML.gif and operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq33_HTML.gif The method used in this paper is different from [17].

Following [13], we say that a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq34_HTML.gif belongs to the function class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq35_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq36_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq37_HTML.gif which satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ8_HTML.gif
(1.8)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq38_HTML.gif and has the partial derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq39_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq40_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq41_HTML.gif is locally integrable with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq42_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq43_HTML.gif

By choosing the special function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq44_HTML.gif it is possible to derive several oscillation criteria for a wide range of differential equations.

Define the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq45_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ9_HTML.gif
(1.9)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq47_HTML.gif The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq48_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ10_HTML.gif
(1.10)
It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq49_HTML.gif is a linear operator and that it satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ11_HTML.gif
(1.11)

2. Main Results

In this section, we give some new oscillation criteria for (1.1). In order to prove our theorems we will need the following lemmas.

Lemma 2.1 (see [5]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq50_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq51_HTML.gif is eventually of one sign for all large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq52_HTML.gif say https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq53_HTML.gif then there exist a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq54_HTML.gif and an integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq55_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq56_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq57_HTML.gif even for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq58_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq59_HTML.gif odd for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq60_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq61_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq62_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq64_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq65_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq66_HTML.gif

Lemma 2.2 (see [5]).

If the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq67_HTML.gif is as in Lemma 2.1 and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq68_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq69_HTML.gif then for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq70_HTML.gif there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq71_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ12_HTML.gif
(2.1)

Lemma 2.3 (see [14]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq72_HTML.gif is an eventually positive solution of (1.1). Then there exists a number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq73_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq74_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ13_HTML.gif
(2.2)

Theorem 2.4.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq75_HTML.gif .

Further, there exist functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq77_HTML.gif , such that for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq78_HTML.gif and for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq79_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ14_HTML.gif
(2.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq80_HTML.gif the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq81_HTML.gif is defined by (1.9), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq82_HTML.gif is defined by (1.10). Then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq83_HTML.gif of (1.1) is oscillatory.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq84_HTML.gif be a nonoscillatory solution of (1.1). Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq85_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq86_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq87_HTML.gif Without loss of generality, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq88_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq89_HTML.gif

By Lemma 2.3, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq90_HTML.gif such like that (2.2) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq91_HTML.gif Using definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq92_HTML.gif and applying (1.1), we get for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq93_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ15_HTML.gif
(2.4)
thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ16_HTML.gif
(2.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq94_HTML.gif

It is easy to check that we can apply Lemma 2.2 for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq95_HTML.gif and conclude that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq97_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ17_HTML.gif
(2.6)
Next, define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ18_HTML.gif
(2.7)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq98_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ19_HTML.gif
(2.8)
From (2.6), (2.7), and (2.8), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ20_HTML.gif
(2.9)
Similarly, define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ21_HTML.gif
(2.10)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq99_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ22_HTML.gif
(2.11)
From (2.6), (2.10), and (2.11), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ23_HTML.gif
(2.12)
Therefore, from (2.9) and (2.12), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ24_HTML.gif
(2.13)
From (2.5), note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq100_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq101_HTML.gif then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ25_HTML.gif
(2.14)
Applying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq102_HTML.gif to (2.14), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ26_HTML.gif
(2.15)
By (1.11) and the above inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ27_HTML.gif
(2.16)
Hence, from (2.16), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ28_HTML.gif
(2.17)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ29_HTML.gif
(2.18)
Taking the super limit in the above inequality, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ30_HTML.gif
(2.19)

which contradicts (2.3). This completes the proof.

We can apply Theorem 2.4 to the second-order neutral delay differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ31_HTML.gif
(2.20)

We get the following new result.

Theorem 2.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq103_HTML.gif Further, there exist functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq105_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ32_HTML.gif
(2.21)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq106_HTML.gif is defined as in Theorem 2.4, the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq107_HTML.gif is defined by (1.9), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq108_HTML.gif is defined by (1.10). Then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq109_HTML.gif of (2.20) is oscillatory.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq110_HTML.gif be a nonoscillatory solution of (2.20). Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq111_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq112_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq113_HTML.gif

Without loss of generality, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq114_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq115_HTML.gif Proceeding as in the proof of Theorem 2.4, we have (2.2) and (2.5) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq116_HTML.gif Next, define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ33_HTML.gif
(2.22)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq117_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ34_HTML.gif
(2.23)
From (2.22) and (2.23), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ35_HTML.gif
(2.24)
Similarly, define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ36_HTML.gif
(2.25)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq118_HTML.gif The rest of the proof is similar to that of the proof of Theorem 2.4, hence we omit the details.

Remark 2.6.

With the different choice of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq119_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq120_HTML.gif Theorem 2.4 (or Theorem 2.5) can be stated with different conditions for oscillation of (1.1) (or (2.20)). For example, if we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq121_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq122_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ37_HTML.gif
(2.26)

By Theorem 2.4 (or Theorem 2.5) we can obtain the oscillation criterion for (1.1) (or (2.20)); the details are left to the reader.

For an application, we give the following example to illustrate the main results.

Example 2.7.

Consider the following equations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ38_HTML.gif
(2.27)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq123_HTML.gif Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq124_HTML.gif it is easy to verify that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_Equ39_HTML.gif
(2.28)

By Theorem 2.5, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq125_HTML.gif one has (2.21). Hence, every solution of (2.27) oscillates. For example, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq126_HTML.gif is an oscillatory solution of (2.27).

Remark 2.8.

The recent results cannot be applied in (1.1) and (2.20) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq127_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq128_HTML.gif Therefore, our results are new.

Remark 2.9.

It would be interesting to find another method to study (1.1) and (2.20) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq129_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq130_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq131_HTML.gif

Remark 2.10.

It would be more interesting to find another method to study (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F184180/MediaObjects/13662_2009_Article_1255_IEq132_HTML.gif is odd.

Declarations

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation funded Project (20080441126, 200902564), and Shandong Postdoctoral funded project (200802018) by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and also by University of Jinan Research Funds for Doctors (B0621, XBS0843).

Authors’ Affiliations

(1)
School of Science, University of Jinan
(2)
School of Control Science and Engineering, Shandong University
(3)
School of Control Science and Engineering, University of Jinan
(4)
Department of Mathematics and Statistics, Missouri University of Science and Technology

References

  1. Hale J: Theory of Functional Differential Equations, Applied Mathematical Sciences, vol. 3. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle Scholar
  2. Džurina J, Stavroulakis IP: Oscillation criteria for second-order delay differential equations. Applied Mathematics and Computation 2003,140(2-3):445-453. 10.1016/S0096-3003(02)00243-6MATHMathSciNetView ArticleGoogle Scholar
  3. Grace SR: Oscillation theorems for nonlinear differential equations of second order. Journal of Mathematical Analysis and Applications 1992,171(1):220-241. 10.1016/0022-247X(92)90386-RMATHMathSciNetView ArticleGoogle Scholar
  4. Koplatadze R: On oscillatory properties of solutions of functional differential equations. Memoirs on Differential Equations and Mathematical Physics 1994, 3: 1-180.MathSciNetGoogle Scholar
  5. Philos ChG: A new criterion for the oscillatory and asymptotic behavior of delay differential equations. Bulletin de l'Académie Polonaise des Science, Série des Sciences Mathématiques 1981, 39: 61-64.Google Scholar
  6. Sun YG, Meng FW: Note on the paper of J. Džurina and I. P. Stavroulakis. Applied Mathematics and Computation 2006,174(2):1634-1641. 10.1016/j.amc.2005.07.008MATHMathSciNetView ArticleGoogle Scholar
  7. Agarwal RP, Grace SR: Oscillation theorems for certain neutral functional-differential equations. Computers & Mathematics with Applications 1999,38(11-12):1-11. 10.1016/S0898-1221(99)00280-1MATHMathSciNetView ArticleGoogle Scholar
  8. Han Z, Li T, Sun S, Sun Y: Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396]. Applied Mathematics and Computation 2010,215(11):3998-4007. 10.1016/j.amc.2009.12.006MATHMathSciNetView ArticleGoogle Scholar
  9. Han Z, Li T, Sun S, Chen W: On the oscillation of second order neutral delay differential equations. Advances in Difference Equations 2010, 2010:-8.Google Scholar
  10. Grammatikopoulos MK, Ladas G, Meimaridou A: Oscillations of second order neutral delay differential equations. Radovi Matematički 1985,1(2):267-274.MATHMathSciNetGoogle Scholar
  11. Karpuz B, Manojlović JV, Öcalan Ö, Shoukaku Y: Oscillation criteria for a class of second-order neutral delay differential equations. Applied Mathematics and Computation 2009,210(2):303-312. 10.1016/j.amc.2008.12.075MATHMathSciNetView ArticleGoogle Scholar
  12. Lin X, Tang XH: Oscillation of solutions of neutral differential equations with a superlinear neutral term. Applied Mathematics Letters 2007,20(9):1016-1022. 10.1016/j.aml.2006.11.006MATHMathSciNetView ArticleGoogle Scholar
  13. Liu L, Bai Y: New oscillation criteria for second-order nonlinear neutral delay differential equations. Journal of Computational and Applied Mathematics 2009,231(2):657-663. 10.1016/j.cam.2009.04.009MATHMathSciNetView ArticleGoogle Scholar
  14. Meng F, Xu R: Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. Applied Mathematics and Computation 2007,190(1):458-464. 10.1016/j.amc.2007.01.040MATHMathSciNetView ArticleGoogle Scholar
  15. Rath RN, Misra N, Padhy LN: Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation. Mathematica Slovaca 2007,57(2):157-170. 10.2478/s12175-007-0006-7MATHMathSciNetView ArticleGoogle Scholar
  16. Şahiner Y: On oscillation of second order neutral type delay differential equations. Applied Mathematics and Computation 2004,150(3):697-706. 10.1016/S0096-3003(03)00300-XMATHMathSciNetView ArticleGoogle Scholar
  17. Xu R, Xia Y: A note on the oscillation of second-order nonlinear neutral functional differential equations. International Journal of Contemporary Mathematical Sciences 2008,3(29–32):1441-1450.MATHMathSciNetGoogle Scholar
  18. Xu R, Meng F: New Kamenev-type oscillation criteria for second order neutral nonlinear differential equations. Applied Mathematics and Computation 2007,188(2):1364-1370. 10.1016/j.amc.2006.11.004MATHMathSciNetView ArticleGoogle Scholar
  19. Xu R, Meng F: Oscillation criteria for second order quasi-linear neutral delay differential equations. Applied Mathematics and Computation 2007,192(1):216-222. 10.1016/j.amc.2007.01.108MATHMathSciNetView ArticleGoogle Scholar
  20. Xu Z, Liu X: Philos-type oscillation criteria for Emden-Fowler neutral delay differential equations. Journal of Computational and Applied Mathematics 2007,206(2):1116-1126. 10.1016/j.cam.2006.09.012MATHMathSciNetView ArticleGoogle Scholar
  21. Ye L, Xu Z: Oscillation criteria for second order quasilinear neutral delay differential equations. Applied Mathematics and Computation 2009,207(2):388-396. 10.1016/j.amc.2008.10.051MATHMathSciNetView ArticleGoogle Scholar
  22. Zafer A: Oscillation criteria for even order neutral differential equations. Applied Mathematics Letters 1998,11(3):21-25. 10.1016/S0893-9659(98)00028-7MATHMathSciNetView ArticleGoogle Scholar
  23. Zhang Q, Yan J, Gao L: Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients. Computers and Mathematics with Applications 2010,59(1):426-430. 10.1016/j.camwa.2009.06.027MATHMathSciNetView ArticleGoogle Scholar

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© Tongxing Li et al. 2010

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